Question

Find the fifth roots of 32(cos 280° + i sin 280°)

Answer 1

`2 ( { \cos \: 56 \degree + i \sin \:56 \degree) }`

Step-by-step explanation

The complex number given to us is in the polar form,

32(cos 280° + i sin 280°)

The fifth root is

`{32}^{ (1)/(5) } ( { \cos280 \degree + i \sin280 \degree) }^{ (1)/(5) }`

This is equal to:

`2 ( { \cos280 \degree + i \sin280 \degree) }^{ (1)/(5) }`

According to the DeMoivre's Theorem,

`( { \cos \theta \: \degree + i \sin\theta \degree) }^{ (p)/(q) } = ( { \cos (p)/(q) \theta \degree + i \sin (p)/(q) \theta \degree) }`

We now use the DeMoivre's Theorem to obtain:

`2 ( { \cos280 \degree + i \sin280 \degree) }^{ (1)/(5) } = 2 ( { \cos \: (1)/(5) * 280 \degree + i \sin \:(1)/(5) * 280 \degree) }`

`2 ( { \cos280 \degree + i \sin280 \degree) }^{ (1)/(5) } = 2 ( { \cos \: 56 \degree + i \sin \:56 \degree) }`